Sunday, May 29, 2011

Initial Conditions and Predictability

According to physicist, Eugene Wigner (article linked in sidebar), physics is possible because we are able to identify regularities in nature.

The world around us is of baffling complexity and the most obvious fact about it is that we cannot predict the future... It is, as Schrodinger has remarked, a miracle that in spite of the baffling complexity of the world, certain regularities in the events could be discovered. One such regularity, discovered by Galileo, is that two rocks, dropped at the same time from the same height, reach the ground at the same time. The laws of nature are concerned with such regularities...

This is surprising, he says, for a few reasons. The first is that it is true everywhere on earth and it will always be true, i.e., it is invariant. The second reason that it is surprising is that this invariance "is independent of so many conditions that could have an effect on it." It doesn't matter where on earth or by whom the rocks are dropped and "there are innumerable other conditions which are all immaterial from the point of view of the validity of Galileo's regularity." He says,

The irrelevancy of so many circumstances which could play a role in the phenomenon observed has also been called an invariance. However, this invariance is of a different character from the preceding one since it cannot be formulated as a general principle. The exploration of the conditions which do, and which do not, influence a phenomenon is part of the early experimental exploration of a field. It is the skill and ingenuity of the experimenter which show him phenomena which depend on a relatively narrow set of relatively easily realizable and reproducible conditions. (Italics added.)

This is important when it comes to understanding the surprising thing about making predictions. Wigner says, "the law of nature is contained in the statement that the length of time which it takes for a heavy object to fall from a given height is independent of the size, material, and shape of the body which drops."

Given certain "easily realizable and reproducible" Initial Conditions (ICs) and a Law of Nature (LN), then a Prediction will follow. (IC & LN) > P. However,

the laws of nature can be used to predict future events only under exceptional circumstances - when all the relevant determinants of the present state of the world are known. It is also in consonance with this that the construction of machines, the functioning of which he can foresee, constitutes the most spectacular accomplishment of the physicist. In these machines, the physicist creates a situation in which all the relevant coordinates are known so that the behavior of the machine can be predicted. Radars and nuclear reactors are examples of such machines.

The two important points I draw from the preceding exposition are:

In order to apply a law to make a prediction you need to identify the relevant conditions (ICs).

The ability to isolate phenomena whose regularity can be demonstrated with easily reproducible conditions cannot be summed up in a general principle.

To those two points I would also add an idea that I think follows from an insight of Stuart Kauffman, author of Reinventing the Sacred (linked on the sidebar) where he says, in analyzing mechanical systems,

physicists since Newton have put in the constraints..."by hand" as what are called the mathematical "boundary conditions" on a system, rather like the boundaries of a billiard table, that keep the balls from rolling off into infinity or under the table. Given the boundary conditions, physicists state the initial conditions, particles and forces, and solve the equations for the subsequent dynamics...

But in the real universe, we can ask, "Where do the constraints themselves come from?"(pp. 90-91) (italics added.)

Although in principle it might be possible to retroactively determine the sequence of events that gave rise to a naturally occurring phenomenon (then again, there's the problem of infinite regress), no one could specify the relevant conditions in advance of every phenomenon-to-be in this complex world. What would we be looking for?

Kauffman discusses the evolution of life and considers Darwinian preadaptations; e.g., three bones in the jaw of an ancestral fish that become bones in the middle ear in species descended from it. At the time this ancient fish lived, was there any way to predict the role of its bones in later species, let alone what later species would emerge?

This is not just an issue of complexity or chaos; i.e., sensitivity to initial conditions (a.k.a "the butterfly effect"). It is that, as Kauffman says, we can't prestate the relevant conditions (p. 139 ff). What potentials will become actualities? In Kauffman's view, this is not just an epistemic limitation, it is an ontological one. He says, "the evolution of life violates no law of physics, but cannot be reduced to physics." Not all of nature is law governed.

So it seems to Kauffman, and to me, that the "natural law" model of a mechanistic universe has limited applicability; i.e., to "phenomena which depend on a relatively narrow set of relatively easily realizable and reproducible conditions."

We cannot completely predict the future, and generally that idea gives me comfort. Nature has space to play.